An Empirical Comparison of Affine and Non-Affine Models for Equity Index Options

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1 An Empirical Comparison of Affine and Non-Affine Models for Equity Index Options Peter Christoffersen Kris Jacobs Karim Mimouni Faculty of Management, McGill University October 21, 25 Abstract The existing literature on equity index option valuation largely focuses on affine models, because they lead to closed-form solutions for option prices. This paper investigates the empirical biases associated with affine models, using data on S&P5 call options. We find that the root mean squared dollar error for a simple non-affine continuous-time stochastic volatility model is 17.4% lower than that of the benchmark continuous-time affine stochastic volatility model in-sample, and almost 2% out-of-sample. The analytical convenience of affine option valuation models therefore comes at a price, and non-affine models ought to be investigated more extensively. We also compare the empirical performance of affine and non-affine discrete-time models. While the performance of the discrete-time non-affine model is similar to that of the continuous-time non-affine model, the discrete-time affine model outperforms the continuous-time affine model. These findings have some interesting implications for the interpretation of existing limit results. At the methodological level, our analysis uses a novel technique based on the Auxiliary Particle Filter. This technique allows for an analysis of option valuation models using options data that imposes consistency with underlying equity returns. It is more straightforward to implement than existing methods and can be used in a variety of applications and on various loss functions. JEL Classification: G12 Keywords: stochastic volatility; GARCH; affine model; option valuation; filtering; out-ofsample; particle filter; continuous-time limit. Christoffersen and Jacobs are also affiliated with CIRANO and CIREQ and want to thank FQRSC, IFM 2 and SSHRC for financial support. Mimouni was supported by a grant from IFM 2. We also thank Jin Duan, Nour Meddahi and Marcel Rindisbacher for helpful comments. Any remaining inadequacies are ours alone. Correspondence to: Peter Christoffersen, Faculty of Management, McGill University, 11 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 1G5; Tel: (514) ; Fax: (514) ; peter.christoffersen@mcgill.ca. 1

2 1 Introduction Following the finding that Black-Scholes (1973) model prices systematically differ from market prices, the literature on option valuation has formulated a number of theoretical models designed to capture these empirical biases. One particularly popular modeling approach has attempted to correct the Black-Scholes biases by modifying the assumption that volatility is constant across maturity and moneyness. Estimates from returns data and options data indicate that return volatility is time-varying, and modeling volatility clustering leads to significant improvements in the performance of option pricing models. It has also been demonstrated that it is necessary to model a leverage effect. The leverage effect captures the negative correlation between returns and volatility, and thus generates negative skewness in the distribution of the underlying asset return. 1 The existing literature has almost exclusively modeled volatility clustering and the leverage effect within an affine structure. Affine models of option valuation are convenient because they lead to closed-form solutions for prices of European equity options. In particular, the affine Heston (1993) model, which accounts for time-varying volatility and a leverage effect, has been implemented in a large number of empirical studies. In order to address the limitations of the affine structure, the Heston (1993) model is often combined with models of jumps in returns and/or volatility. 2 However, relatively little is known about the empirical biases that result from imposing the affine structure. 3 The existing literature has also almost exclusively modeled volatility clustering and the leverage effect using continuous-time stochastic volatility models. There exists a discrete-time literature on option valuation using GARCH processes, and in this literature the distinction between affine and non-affine models is also relevant. The affine GARCH dynamic in Heston and Nandi (2) yields a closed-form solution for option prices, but the extensive empirical literature on GARCH processes that concerns itself with fitting and predicting return volatility almost exclusively uses non-affine processes. 4 A small number of studies investigate option valuation assuming non-affine GARCH 1 The leverage effect was first characterized in Black (1976). For empirical studies that emphasize the importance of volatility clustering and the leverage effect for option valuation see among others Benzoni (1998), Chernov and Ghysels (2), Eraker (24), Heston and Nandi (2), Nandi (1998) and Pan (22). 2 For empirical studies that implement the Heston (1993) model by itself or in combination with different types of jump processes, see for example Andersen, Benzoni and Lund (22), Bakshi, Cao and Chen (1997), Bates (1996, 2), Benzoni (22), Chernov and Ghysels (2), Huang and Wu (24), Pan (22), Eraker (24) and Eraker, Johannes and Polson (23). 3 Jones (23) and Benzoni (22) are notable exceptions that investigate non-affine option valuation models. The non-affine model in Benzoni (22) does not improve on the performance of the Heston (1993) model, while a number of specification tests in Jones (23) favor the non-affine constant elasticity of substitution model over the Heston (1993) model. 4 The literature on GARCH processes is too voluminous to cite in full here. The classical references are Engle (1982) and Bollerslev (1986). See Bollerslev, Chou and Kroner (1992) and Diebold and Lopez (1995) for reviews. 2

3 processes for the underlying securities. 5 This paper investigates the empirical implications of adopting an affine framework for option valuation. We compare the empirical performance of the affine Heston (1993) stochastic volatility model (AF-SV) with that of a simple non-affine stochastic volatility model (NA-SV). We estimate model parameters using a long time series of cross sections of options data in a framework that imposes consistency with the underlying equity returns. We conduct this empirical analysis using a novel setup that uses the Auxiliary Particle Filter algorithm. This methodology provides a convenient filtering algorithm for latent factor models such as stochastic volatility models. Our new methodology is relatively easy to implement compared with existing likelihood-based methods, and it can be adapted to provide the best possible fit to the objective function of interest. 6 We also compare the empirical performance of the affine Heston and Nandi (2) model (AF-GARCH) with that of the non-affine GARCH model (NA-GARCH) of Engle and Ng (1993). We find that the affine framework is very restrictive. We conduct six in-sample exercises and six out-of-sample exercises, and despite the fact that both the AF-SV and the NA-SV models show signs of misspecification in several samples, the NA-SV model outperforms the Heston (1993) AF- SV model in all of these exercises. The NA-SV model also outperforms the AF-SV model for all moneyness and maturity categories. On average, the dollar RMSE of the NA-SV we investigate is approximately 17.4% lower than that of the AF-SV model in-sample, and the out-of-sample dollar RMSE is almost 2% lower. We therefore conclude that while the closed-form solution provided by affine models is convenient, this analytical convenience comes at a price, and non-affine models need to be studied more extensively. Interestingly, the differences in RMSE between the non-affine discrete-time model (NA-GARCH) and the affine discrete-time model (AF-GARCH) are smaller than in the continuous-time case, approximately 7.5% in-sample and 1% out-of-sample. 7 This finding naturally raises important questions about the relationship between the empirical performance of discrete-time and continuoustime models. Our findings on this issue are perhaps less straightforward to interpret than those on the implications of the affine structure, but we are able to draw a number of important conclusions. The literature contains a number of limit results relating certain classes of discrete-time and continuous-time models (see for example Duan (1997), Heston and Nandi (2) and Nelson (199)). Some researchers have interpreted these limit results as evidence that the performance of discretetime and continuous-time models ought to be very similar when the continuous-time dynamic is the 5 See Amin and Ng (1993), Bollerslev and Mikkelsen (1996), Engle and Mustafa (1992), Heston and Nandi (2), Christoffersen and Jacobs (24) and Duan, Ritchken and Sun (22). 6 Likelihood-based approaches fully exploit the available information, but do so at the cost of greater complexity. See Eraker (24) and Jones (23) for Bayesian approaches. likelihood approach that is relatively easier to implement. 7 See also Hsieh and Ritchken (2) for a related comparison. See Bates (24) for an approximate maximum 3

4 limit of the discrete-time dynamic. This interpretation is somewhat contentious, because a given discrete-time model can have several continuous-time limits, and a given continuous-time model can be the limit for more than one discrete-time model (see for instance Corradi (2)). While limit results are therefore theoretically intriguing, in some cases their practical relevance may be limited. We find some very interesting differences in the empirical performance of models that are often thought of as equivalent. Most importantly, the AF-SV Heston (1993) model, which is the benchmark model in the continuous-time stochastic volatility literature, significantly underperforms the AF-GARCH Heston-Nandi (2) model. This result is perhaps surprising, because Heston and Nandi (2) demonstrate that a restricted Heston (1993) model can be seen as the limit of the AF-GARCH model, but the key is of course that this particular limit is just one of the many available. Which limit obtains depends on the particular mathematical construction used. This finding does not mean that discrete-time GARCH models outperform continuous-time stochastic volatility models. Indeed, the performance of the non-affine NA-GARCH discrete time model of Engle and Ng (1993) is very similar to the performance of the NA-SV model we investigate. These findings demonstrate that it is difficult to make general statements about discrete-time and continuous-time models. Certain stylized facts may be more conveniently captured by specific continuous-time models, while others are more easily modeled using a particular discrete-time model. This paper merely provides a start to that discussion by documenting the empirical performance of some important benchmark models. After documenting that the benchmark AF-SV Heston (1993) model violates our prior by underperforming a related discrete-time model, we proceed by suggesting an aspect of the model that may cause the underperformance. One final remark is in order. Traditionally, the existence of multiple limits has not been the only issue that complicated a comparison of discrete-time and continuous-time models. The two classes of models are typically implemented using very different econometric methods, which renders fair comparisons difficult. Our use of the Auxiliary Particle Filter algorithm allows for straightforward comparisons of latent factor volatility continuous time models with discrete time GARCH models, because each model is implemented using the same objective function and the same information set. The novel estimation setup in this paper therefore facilitates comparisons between different classes of models. The paper proceeds as follows. In Section 2 we introduce the discrete- and continuous-time volatility models, and we discuss their implementation. In Section 3 we present and discuss the empirical results. Section 4 concludes. 4

5 2 Volatility Dynamics and Model Implementation We now turn to a description of the four volatility models we estimate and we discuss their implementation for the purpose of option valuation. We first discuss our implementation of the Heston (1993) model, which is different from the implementation available in the literature. Our implementation is designed to provide the best possible fit for the model, as well as to facilitate the comparison with the discrete-time models. Interestingly, this implementation is relatively straightforward and substantially simpler compared to other methods that analyze options data while imposing consistency with underlying returns data. Subsequently, we discuss the specification of a non-affine stochastic volatility model, and finally we discuss the two discrete-time GARCH models, where we consider the GARCH(1,1) representation because it is most closely related to the Heston (1993) continuous-time model. Following a discussion of the implementation of the discrete-time Heston and Nandi (2) model, we conclude with the specification of a non-affine discrete-time GARCH model. The Heston (1993) model is arguably the most popular model in the index option valuation literature. Its closed-form solution is due to the model s affine structure. 8 Because the existing literature does not contain strong evidence on modifications of the stochastic volatility dynamic that outperform the Heston (1993) model out-of-sample for index options, it constitutes a good benchmark for our study. 9 The underlying reason for its success is that the Heston (1993) model captures two important stylized facts that are needed to model option prices: volatility clustering and the leverage effect. After accounting for these two stylized facts, additional modifications of the return and volatility dynamic do not seem to result in significant out-of-sample improvements in the fit. We refer to the affine Heston (1993) model as AF-SV below. The second model we investigate has not been extensively analyzed in the literature. It is a continuous time non-affine stochastic volatility model and we refer to it as NA-SV below. It contains a latent factor volatility which is correlated with returns, but it does not allow for closedform option valuation. Consequently it is more complex to implement than the Heston (1993) model. The discrete-time GARCH option valuation literature has resulted in relatively few empirical studies, 1 while discrete-time volatility modeling using only returns data has spawned a large num- 8 Affine models are also very popular in the term structure literature for exactly the same reason. See for instance Duffie and Kan (1996) and Dai and Singleton (2). 9 There is an extensive and growing literature on the use of jumps in returns and volatility to improve the performance of the Heston model. See Andersen, Benzoni and Lund (22), Bakshi, Cao and Chen (1997), Bates (1996, 2), Chernov, Gallant, Ghysels and Tauchen (23), Eraker, Johannes and Polson (23), Eraker (24), Pan (22), Broadie, Chernov and Johannes (24), Carr and Wu (24) and Huang and Wu (24). Extending our comparison to models of this type is interesting, but beyond the scope of this paper. 1 Duan (1995) and Amin and Ng (1993) provide theoretical foundations for this literature. Bollerslev and Mikkelsen 5

6 ber of competing models following the work of Engle (1982) and Bollerslev (1986). Our benchmark discrete-time specification is the model of Heston and Nandi (2), which was designed with option valuation in mind. Like the Heston (1993) model, it contains a leverage effect, it allows for volatility clustering, and it leads to a closed-form solution due to its affine structure. Heston and Nandi (2) have demonstrated that this model performs satisfactorily vis-a-vis ad-hoc benchmarks for the purpose of option valuation. We refer to this model as AF-GARCH. The other discrete-time model we investigate is the non-affine NGARCH model of Engle and Ng (1993), henceforth referred to as NA-GARCH. This is the simplest model in the GARCH literature that contains both volatility clustering and a leverage effect, and was first considered for option valuation by Duan (1995). Christoffersen and Jacobs (24a) demonstrate that several richer GARCH parameterizations do not improve on the option valuation performance of the NA- GARCH model. This model does not lead to a closed-form solution for option prices, and has to be analyzed using numerical methods. Because the NA-GARCH dynamic has proven extremely valuable in modeling equity returns as well as other financial time series, it is of interest to verify whether the focus on closed-from valuation results in the options literature comes at the cost of a deterioration in the model s empirical performance. We now give the specifics on each of the four models as well as a detailed description of their empirical implementation, which imposes consistency between option prices and underlying index returns. 2.1 The Affine Stochastic Volatility Model (AF-SV) The Heston (1993) continuous-time stochastic volatility model (AF-SV) is definedbythefollowing two equations ds = µsdt + VSdw S (1) dv = κ(θ V )dt + σ Vdw V (2) with corr(dw S,dw V )=ρ. This model allows for volatility clustering through the autoregressive component of volatility, and for a leverage effect through a negative correlation coefficient ρ, which translates into negative skewness of the return distribution. 11 Under the assumption that the volatility risk premium λ(s, V, t) is equal to λv, the risk neutral dynamic expressed in terms of the (1996), Engle and Mustafa (1992), and Duan, Ritchken and Sun (22) estimate model parameters using the underlying asset returns and subsequently value options. Heston and Nandi (2) and Christoffersen and Jacobs (24a) estimate model parameters using equity option prices, and impose consistency with the underlying returns. 11 Note that following Chernov and Ghysels (2), Eraker, Johannes and Polson (23) and Eraker (24) we use a simple constant specification for the stock return drift. 6

7 physical parameters is ds = rsdt + VSdw S (3) dv = (κ(θ V )+λv ) dt + σ Vdw V (4) = (κ λ)(κθ/(κ λ) V )dt + σ Vdw V (5) with corr(dw S,dw V )=ρ. Heston (1993) demonstrates that this model admits a closed form solution for option prices, which is presented here in terms of the physical parameters κ, θ, λ, ρ and σ in order to facilitate the description of our estimation procedure below. C(V )=SP 1 Ke r(t t) P 2 (6) where and P j = π Z exp( iφ ln(k))fj (ln(s),v,t; φ)) Re dφ, j =1, 2 iφ f j (ln(s),v,t; φ) = exp(c(t t; φ)+d(t t; φ)v + iφ ln(s)) C(τ; φ) = rφiτ + κθ µ 1 g exp(dτ) σ 2 (b j ρσφi + d)τ 2ln 1 g D(τ; φ) = b µ j ρσφi + d 1 exp(dτ) σ 2 1 g exp(dτ) d = q (ρσφi b j ) 2 σ 2 (2µ j φi φ 2 ) g = b j ρσφi + d b j ρσφi d µ 1 = 1 2, µ 2 = 1 2, b 1 = κ λ ρσ, b 2 = κ λ The Heston model has been investigated empirically in a large number of studies. Often it is used as a building block together with models of jumps in return and volatility. For our purpose, it is important to note that the model can be estimated and investigated empirically using a number of different techniques. First, the model s parameters can be estimated using a single cross-section of option prices (for example see Bakshi, Cao and Chen (1997)). A second type of implementation of the Heston model uses multiple cross sections of option prices but does not use the information in the underlying asset returns. Instead, for every cross section a different initial volatility is estimated, leading to a highly parameterized problem (see for instance Bates (2) and Huang and Wu (24)). A third group of papers provide a likelihood-based analysis of the stochastic volatility model. For instance, Eraker (24) provides a Markov Chain Monte Carlo analysis. Finally, Chernov and Ghysels (2) use the efficient method of moments and Pan (22) uses a 7

8 method of moments technique as well. These methods can also combine information in the options data as well as the underlying returns. In this paper we implement the Heston model in a novel way. This is mainly motivated by our desire to compare the performance of discrete-time and continuous-time methods in a meaningful way. Our implementation uses nonlinear least squares (NLS) estimation techniques to minimize $MSE = 1 X N T (C i,t C i (V )) 2 (7) t,i with respect to the physical parameters µ, κ, θ, λ, ρ and σ, wheren T P = T N t, T is the total number of days included in the options sample, N t is the number of options with various strikes prices and maturities included in the sample at date t, C i,t is the market price of option i quoted on day t and C i (V ) is the model price. In our opinion, this type of objective function guarantees the best possible performance for the model in- and out-of-sample. This is motivated by the insights of Granger (1969), Weiss (1996) and Weiss and Andersen (1984) who demonstrate that the choice of objective function (also labeled loss function) is an integral part of model specification. It follows that estimating a model using one objective function and evaluating it using another one amounts to suboptimal choice of objective function. Christoffersen and Jacobs (24b) demonstrate that this issue is empirically relevant for the estimation of the deterministic volatility functions in Dumas, Fleming and Whaley (1998). We therefore implement the Heston model in a way that is consistent with these insights. The problem with (7) is that V is unobservable, and that it has to be filtered from observed data. Filtering the latent volatility factor on observed index returns as we do below avoids overfitting and ensures that the option valuation model is consistent with both options and returns data. Our implementation uses the Auxiliary Particle Filter (APF) algorithm. 12 AsshownbyPittand Shephard (1999) the APF offers a convenient filtering algorithm for non-linear models such as the stochastic volatility model we consider here. Because the APF procedure is relatively new in finance, we now discuss the implementation of this method in more detail. 13 t= Volatility Transformation and Discretization To prevent V from becoming negative, we work with f(v )=ln(v). Using Ito s lemma, the dynamic of interest is therefore d ln(v )= 1 µκ(θ V ) 12 V σ2 dt + σ 1 dw V (8) V 12 We have also implemented the Sampling-Importance-Resampling (SIR) particle filter as a robustness check, and this yields similar results. 13 Johannes, Polson and Stroud (22) discuss the use of the particle filter to estimate parameters for continuoustime jump-diffusion models on returns data. 8

9 Note that equations (1) and (2) specify how the unobserved state is linked to observed stock prices. This relationship allows us to infer the volatility path using the returns data. We first need to discretize equations (1) and (8). There are different discretization methods and every scheme has certain advantages and drawbacks. We use the Euler scheme which is easy to implement and has been found to work well for this type of applications. Discretizing equation (1) (using ln(s) instead of S and applying Ito s lemma again) and (8) gives µ ln(s t+1 ) = ln(s t )+ µ 1 2 V t + p V t ε S t+1 (9) ln(v t+1 ) = ln(v t )+ 1 µ κ (θ V t ) 1 V t 2 σ2 + σ 1 ε V t+1 (1) Vt We implement the discretized model in (9) and (1) using daily returns, and all parameters will be expressed in daily units below. The model is characterized by six structural parameters: µ, κ, θ, σ, λ and ρ for which we have to choose a set of starting values. Subsequently, we have to choose an initial variance V (the starting value for the variance path). We set the initial variance equal to the model-implied unconditional variance, V = θ. Our optimization algorithm minimizes (7) using an iterative procedure. At each iteration, the volatility is filtered using the information embedded in observed returns. Since the minimization is performed relative to option prices, option data also indirectly contribute to the determination of the volatility path. Finally, using the filtered volatility and the structural parameters option prices are computed according to Heston s formula and the MSE is calculated. This procedure is repeated until the optimum is reached. We now describe volatility filtering in more detail Filtering the volatility path using the APF algorithm The idea underlying the AP F technique is to infer the volatility path from the observed returns data. V t is propagated one day ahead using equation (8) into N possible states (or particles). 14 Subsequently, we use an auxiliary variable ι and the available data to decide which particles to keep in order to simulate the one day ahead volatility. Assume that we are at date t and we have an initial set of particles n V j t,wj t o N t j=1 the volatility at day t for the state j, W j t the weight associated with state j at date t and j = n o N 1,..,N. 15 We want to propagate V t one day ahead into V j t+1,wj t+1 requires the following steps: using the AP F.Thistask j=1 Step 1: Selecting the particles 14 We set N =5in the initial search. Once a candidate optimum is identified we confirm it by increasing N to 5,. The results change very little when N is increased. 15 At time, the initial set is constructed by setting each particle equal to the unconditional variance θ and giving all particles equal weight, 1/N. 9

10 The weight W j t reflects information available at time t only and does not include our expectations about (t +1). So even if state j is very likely according to the realization of the stock price at date t, it is possible that the realization of the stock at date (t +1) suggests a certain re-adjustment of the probability that state j has occurred. By combining the information available at t and our expectations about (t +1), we can eliminate many states with a low realization probability right before the propagation step. This is achieved by: I) Computing a summary location statistic for (t +1) that reflects the information at t. Weuse the mean µ j t+1 given by: ³ µ j t+1 = E ln(v t+1 ) V j t II) Simulating the auxiliary variable ι j ι j W j t ³ln(S p t+1 ) µ j t+1 ³ where p ln(s t+1 ) µ j t+1 is the conditional density of ln(s t+1 ) which can be easily inferred from (9). This auxiliary variable is simply an index that tells us which particle to keep and which particle to discard. After this selection exercise we obtain N new particles which are implicitly functions of the auxiliary variable ι, {V (ι) t,w(ι) t } N j=1 In order to keep the notation simple we will omit ι below. Step 2: Simulating the state forward (Sampling) This is done by computing V t+1 using equation (1) and taking the correlation into account. We have µ µ St+1 ln = µ 1 2 V t + p V t ε S t+1 which gives Since where corr(ε S t+1,ε t+1) =,weget S t ³ ln St+1 ε S S t µ 1 2 V t t+1 = Vt ε V t+1 = ρε S t+1 + p 1 ρ 2 ε t+1 ln(v t+1 )= ³ ln(v t )+ 1 µ κ (θ V t ) 1 V t 2 σ2 + σ 1 ln St+1 S t µ 1 2 ρ V t + p 1 ρ 2 ε t+1 Vt Vt We simulate N states (N particles) which describe the set of possible values of V t+1. Step 3: Computing and normalizing the weights (Importance Sampling) 1

11 At this point, we have a vector of N possible values of V t+1 and we know according to equation (9) that given the other available information, V t+1 is sufficient to generate ln(s t+2 ). Therefore, equation (9) offers a simple way to evaluate the likelihood that the observation S t+2 has been generated by V t+1. Hence, we have to compute the vector W whose elements represent the weight given to each particle (or the likelihood or probability that the particle j has generated S t+2 ). The likelihood is computed as follows: ³ ³ ³ W j t+1 = 1 q exp 1 ln St+2 S t+1 µ 1 2 V j 2 t+1 V j 2 V j t+1 t+1 This is to be repeated for j =1,..,N. Finally, because nothing guarantees that P N j=1 W j t+1 =1, we have to normalize and set W j t+1 = W j t+1 P Nj=1. In summary therefore, at the end of Step 2, we W j t+1 obtain a set of N particles describing the density of V t+1. This procedure (Steps 1, 2 and 3) is repeated for t =1,...T. To obtain the filtered volatility path, we then compute for each t. V t+1 = NX W j t+1v j t+1 j= Computing option prices and evaluating the loss function We are now in a position to evaluate option prices C i Vt based on the filtered volatility path using Heston s closed form solution according to equation (6). We subsequently evaluate the loss function (7), using C i Vt for the model price Ci (V ). We use a standard numerical optimization routine to update the model parameters and iterate until convergence is achieved. 16 Notice that the methodology we have suggested here for estimating the continuous time stochastic volatility model relies on the same information set and uses the same objective function as the two discrete time models, which will be discussed in detail below. This will allow for a fair empirical comparison between models. 2.2 The Non-Affine Stochastic Volatility Model (NA-SV) While the empirical literature does not offer much guidance on the choice of a non-affine stochastic volatility models, and several sensible specifications are available, we focus on a process which has the same number of parameters as the affine SV model above. Our non-affine model is inspired by 16 In order to mitigate the impact of the choice of the initial variance V we start iterating on the volatility dynamic on January 2, 1989 in all the estimation samples and for all the models. 11

12 the properties of the NGARCH model of Engle and Ng (1993), 17 which we use later in this paper as our NA-GARCH specification. We use the following NA-SV dynamic ds = µsdt + VSdw S (11) dv = κ(θ V )dt + σv dw V (12) with corr(dw S,dw V )=ρ. In this model the innovations are scaled by the conditional variance rather than by the square root of the conditional variance as is the case in the AF-SV model. It must also be noted that our NA-SV model is related to the Engle and Ng (1993) model in a more precise way, because it can be shown to be the limit of the Engle and Ng (1993) model, when the limit is defined according to Duan (1997). Under the assumption that the volatility risk premium λ(s, V, t) is equal to λv, the risk neutral dynamic expressed in terms of the physical parameters is ds = rsdt + VSdw S (13) dv = (κ(θ V )+λv ) dt + σv dw V (14) = (κ λ)(κθ/(κ λ) V )dt + σv dw V (15) with corr(dw S,dw V )=ρ. Note that the assumption on the volatility risk premium is the same as in the AF-SV model. In both cases the risk-neutralization can be obtained using a no-arbitrage argument, but in the AF-SV case the risk-neutral dynamic can also be obtained using a utility-based argument. For the NA-SV model, a utility-based risk-neutralization may be possible, but this cannot be done under the assumption that λ(s, V, t) is equal to λv. If we change both the variance dynamic and the price of risk, the resulting change in model fit can be due to either assumption. In order to isolate the importance of the volatility dynamic, we keep the volatility risk premium specifications constant across the affine and non-affine SV models. 18 The NA-SV model presented here has an unobserved variance factor and no closed form option valuation formula. Thus we need to implement it using the auxiliary particle filter to construct the variance path (as in AF-SV) and Monte Carlo simulation to calculate the option prices. We use 1 simulated paths and a number of numerical techniques to increase numerical efficiency: the empirical martingale method of Duan and Simonato (1999), stratified random numbers, antithetic variates and a control variate technique. The model is estimated by minimizing (7) with respect to the physical parameters µ, κ, θ, λ, ρ and σ. 17 See also Duan (1997) and Ritchken and Trevor (1999). 18 See Lewis (2) for a thorough discussion of these issues. 12

13 2.3 The Affine GARCH(1,1) Model (AF-GARCH) Heston and Nandi (2) propose a class of affine GARCH models (AF-GARCH) that allow for a closed-form solution for the price of a European call option. We investigate the GARCH(1,1) version of this model, which is given by ln(s t+1 ) = ln(s t )+r + ph t+1 + p h t+1 z t+1 (16) h t+1 = w + bh t + a ³z t c p h t 2 (17) where S t+1 denotes the underlying asset price, r the risk free rate, p the price of risk and h t+1 the daily variance on day t +1which is known at the end of day t. The z t+1 shock is assumed to be i.i.d. N(, 1). The Heston-Nandi model captures time variation in the conditional variance in ways similar to Engle (1982) and Bollerslev (1986). The parameter c represents the leverage effect, which captures the negative relationship between returns and volatility (Black (1976)) and results in a negatively skewed conditional distribution of multi-day returns. Note also that using the conventional GARCH notation, the conditional variance for day t +1, denoted h t+1,isknown at the end of day t. Variance persistence can be computed via b + ac 2 1 κ and the unconditional variance can be computed via (w + a)/ 1 b ac 2 =(w + a)/κ θ Now we can rewrite the variance process as ³ z 2 h t+1 h t = κ(θ h t )+a t 1 p 2cz t ht (18) which suggests the model s relationship with the diffusion volatility models considered above. The risk-neutral dynamics for the GARCH(1,1) model (16)-(17) are given by 19 ln(s t+1 ) = ln(s t )+r 1 2 h t+1 + p h t+1 zt+1 (19) h t+1 = w + bh t + a(zt (c + p +.5) p h t ) 2 with zt N(, 1) under the risk neutral measure. We provide an analysis of this model using data on equityoptionpricesaswellasthetimeseries of underlying equity returns. In order to value options at each date t, we need an estimate of the conditional volatility h t+1 on that particular date. One of the appealing aspects of discrete-time 19 For the underlying theory on risk neutral distributions in discrete time option valuation see Rubinstein (1976), Brennan (1979), Amin and Ng (1993), Duan (1995), Camara (23), Heston and Nandi (2) and Schroder (24). 13

14 GARCHmodelsisthatthisfiltering problem is extremely simple. Indeed, the filtering problem is solved by noting that from (16) we have z t+1 =(R t+1 r ph t+1 ) / p h t+1 (2) where R t =ln(s t /S t 1 ). Substituting (2) in (17), it can be seen that the updating from h t to h t+1 is done using an updating function that exclusively involves observables h t+1 = w + bh t + a((r t r) / p h t (c + p) p h t ) 2 (21) Model parameters are obtained by using the nonlinear least squares (NLS) estimation techniques to minimize $MSE = 1 X (C i,t C i (h t+1 )) 2 (22) NT t,i P where NT = T N t, T is the total number of days included in the options sample and N t is the t=1 number of options included in the sample at date t, C i,t is the market price of option i quoted on day t and C i (h t+1 ) is the model price. The implementation is therefore relatively simple: the NLS routine is called with a set of parameter starting values. The variance dynamic in (21) is then used to update the variance from day to day and the GARCH(1,1) option valuation formula from Heston and Nandi (2) is used to compute the model prices. At time t, a European call option with strike price K that expires at time T can be calculated from where C (h t+1 )=e r(t t) E t [Max(S T K, )] (23) P 1 = π P 2 = π = S t P 1 Ke r(t t) P 2 Z Z K iφ f (t, T ; iφ +1) Re iφs t er(t t) K iφ f (t, T ; iφ) Re dφ iφ dφ (24) and where f (t, T ; iφ) is the conditional characteristic function of the logarithm of the spot price under the risk neutral measure, which is characterized by a set of difference equations with terminal conditions. See Heston and Nandi (2) for these equations. 2.4 The Non-Affine GARCH Model (NA-GARCH) We compare the AF-GARCH model with the non-affine NGARCH model of Engle and Ng (1993). We choose the NGARCH model because it is relatively easy to analyze, and because it has been 14

15 shown to provide a good description of underlying equity returns. We will refer to this model as NA-GARCH. The model is given by ln(s t+1 ) = ln(s t )+r + p p h t+1.5h t+1 + p h t+1 z t+1 (25) h t+1 = w + bh t + ah t (z t c) 2 (26) The variance persistence can be computed via b + a 1+c 2 1 κ and the unconditional variance can be computed via w/ 1 b a 1+c 2 = w/κ θ Note we can rewrite the variance process as h t+1 h t = κ(θ h t )+ah t z 2 t 1 2cz t (27) Notice that the variance of the shock term is 2+4c 2 so that we have h t+1 h t = κ(θ h t )+σh t v t where σ = a 2+4c 2 and v t = z 2 t 1 2cz t / 2+4c 2.Noticealsothatwehave Corr(z t,v t )= 2c 2+4c 2 ρ all of which suggests the NA-GARCH model s close relationship with the NA-SV model. Note that this model differs in some subtle ways from the Heston-Nandi model in (16)-(17). The Heston-Nandi model was engineered with the specific purpose of yielding closed-from option prices. The specification in (25)-(26) does not yield closed form option prices, but was designed to provide a good fit to the underlying equity returns. The question of interest is if the restrictions built into affine models such as (16)-(17) reduce the ability of the model to fit the data. The risk-neutral dynamics for the NA-GARCH model (25)-(26) can be obtained using the same theoretical arguments underlying the Heston-Nandi model ln(s t+1 ) = ln(s t )+r.5h t+1 + p h t+1 zt+1 (28) h t+1 = w + bh t + ah t (zt (c + p)) 2 with zt N(, 1). We can then estimate the model by minimizing (22), using the updating rule ³h h t+1 = w + bh t + ah t (R t r +.5h t 1 ) / p i 2 h t 1 (c + p) (29) 15

16 Option prices are computed numerically according to C (h t+1 )=e r(t t) Et [Max(S T K, )] where the expectation is calculated by Monte Carlo simulation of the daily returns from (28). We use the same Monte-Carlo setup as in the NA-SV case, with 1 simulated paths and a number of numerical techniques to increase numerical efficiency: the empirical martingale method of Duan and Simonato (1999), stratified random numbers, antithetic variates and a control variate technique. 3 Empirical Results This section presents the empirical results. We first discuss the data, followed by an empirical evaluation of the four models under investigation and a detailed discussion of the differences in performance of these models in- and out-of-sample. 3.1 Data We conduct our empirical analysis using six years of data on S&P 5 call options, for the period We apply standard filters to the data following Bakshi, Cao and Chen (1997). We only use Wednesday and Thursday options data. For the in-sample analysis, we use the Wednesday data. Wednesday is the day of the week least likely to be a holiday. It is also less likely than other days such as Monday and Friday to be affected by day-of-the-week effects. The decision to pick one day every week is to some extent motivated by computational constraints. The optimization problems are fairly time-intensive, and limiting the number of options reduces the computational burden. Using only Wednesday data allows us to study a fairly long time-series, which is useful considering the highly persistent volatility processes. An additional motivation for only using Wednesday data is that following the work of Dumas, Fleming and Whaley (1998), several studies have used this setup (see for instance Heston and Nandi (2)). We obtain six sets of parameter estimates in the in-sample analysis. We simply split the six years of data in six datasets, one for each calendar year, and perform annual estimation exercises. For each estimation sample, we use a volatility updating rule starting from the model implied unconditional variance on January 1, Table 1 presents descriptive statistics for the options data for the Wednesday insample data by moneyness and maturity. Panels A and B indicate that the data are standard. Panel C displays the volatility smirk in the data. The slope of the smirk clearly differs across maturities. We summarize the data for all six estimation samples in one set of tables to save space. Descriptive statistics for the yearly samples (not reported here) reveal similar stylized facts. The slope of the smirk changes over time, but the smirk is present throughout the sample. The top 16

17 panel of Figure 1 gives some indication of the pattern of implied volatility over time. For the 313 Wednesdays of options data used in the empirical analysis, we present the average implied volatility of the options on each Wednesday. It is evident from Figure 1 that there is substantial clustering in implied volatilities. It can also be seen that volatility is higher in the early part of the sample. The bottom panel of Figure 1 presents a time series for the 3-day at-the-money volatility (VIX) index from the CBOE for our sample period. A comparison with the top panel clearly indicates that the options data in our sample are representative of market conditions, although the time series based on our sample is of course a bit more noisy due to the presence of options with different moneyness and maturities. After conducting six in-sample estimations, we proceed to conduct separate out-of-sample analyses for each of the six sample years using the trading day following each in-sample Wednesday. We refer to these datasets as the Thursday data. Table 2 presents descriptive statistics for the out-ofsample data. The patterns in the data are clearly similar to those in the in-sample data in Table Parameter Estimates and Option Mean-Squared-Errors Table 3 presents the parameter estimates for each of the four models and for each of the six annual estimation samples. The parameters for the AF-SV and NA-SV models are directly interpretable individually; for example, κ denotes the daily variance mean reversion and θ denotes the daily unconditional variance. These parameters appear to be relatively stable over time, even though the parameter estimates for 199 and 1991 seem to be somewhat different from the estimates for the subsequent four years. The most striking aspect for the 199 and 1991 estimates is that the correlation parameter ρ hits the prespecified boundary of for the AF-SV and NA-SV models. This seems to indicate that it is difficult for the AF-SV and NA-SV models to match the negative correlation in the data for this period. The NA-SV model also hits the boundary for ρ in 1993 and For the AF-GARCH and NA-GARCH models, we report the parameters from the specifications in (18) and (27) in order to facilitate comparision with the SV models. For the non-affine models the parameters are quite comparable across models. For the affine models, a comparison is less straightforward, partly because the conditional correlation is time-varying in the AF-GARCH model. Note that c, the parameter that determines the size of the leverage effect in the AF-GARCH model, is higher in 199 and 1991, consistent with the ρ estimates for the AF-SV model. To further facilitate the comparison between the different models, Table 4 complements Table 3 by focusing on variance persistence and the unconditional volatility reported under both the physical and risk neutral measures. The variance persistence is close to one for all models, which is consistent with other findings in the literature. It is higher under the risk neutral measure, 17

18 which indicates a negative price of volatility risk. Often the persistence is largest for the AF-SV specification. The unconditional volatility displays considerable variation over time. In 1991 the persistence in the NA-GARCH model is very close to 1, leading to an unrealistically large estimate of the unconditional variance. 2 Keeping in mind the difficulties of the AF-SV and NA-SV models to capture the leverage effect mentioned above, we therefore conclude that it is challenging for most models to provide a satisfactory fit to the 1991 data. The in- and out-of-sample RMSEs from the four models are reported in Table 5 for each of the six samples. First note that the NA-SV model is best overall and the AF-SV model is worst overall both in- and out-of-sample. The overall difference between these models is 17.4% in-sample and 19.8% out of sample. The performance of the NA-GARCH and NA-SV models is similar in- and out-ofsample with the NA-SV model performing somewhat better overall. This correspondence between the fit of the non-affine continuous-time and discrete-time models does not obtain for the affine models. The fit of the AF-GARCH model falls in between the AF-SV and the non-affine models. The AF-GARCH is approximately 7.3% better than the AF-SV in sample and approximately 8.1% better out of sample. Looking across the six samples, a very robust conclusion obtains: the nonaffine models substantially outperform the affine models in every year, both in- and out-of-sample. Nevertheless, we note that the parameter estimates indicate potential misspecification of the NA- SV model. In particular, the estimate of ρ is equal to the pre-specified boundary of in several samples. This observation confirms that further work on the non-affine model is needed. The objective of this paper is not to findthebestpossiblenon-affine model. We merely want to demonstrate that a simple (and possibly misspecified) non-affine model improves on the benchmark affine model. Figure 2 provides further perspective on the similarities and differences between the four models by providing volatility sample paths for the models. The figure plots volatility paths for the four models using two sets of parameters estimates for each, the estimates for the 199 sample (left column) and the estimates for the 1995 sample (right column). It can be seen that despite the fact that there are significant differences in the estimated long-run volatility between the model estimates for the 199 sample, the sample paths for the four models are quite similar. For the 1995 estimates, the models all display increases in volatility around the same period, but the overall sample paths are rather different for the four models. When comparing the sample paths for the 199 and 1995 estimates for a given model, it is clear that the 1995 estimates display lower persistence. The larger volatility of volatility estimates in the 1995 sample for all models except the AF-SV are also apparent. 2 The high unconditional variance does not signal model misspecification because the conditional variance is a function of the product of the unconditional variance and one minus the persistence. 18

19 3.3 Pricing Errors Over Time and Across Moneyness and Maturity We now analyze the performance of the four models in more detail. Figure 3 addresses the performance of the models over time. The top four panels present the RMSE on a week-by-week basis. It can clearly be seen that the four models display important similarities in terms of the pricing patterns that they can and cannot explain. This observation is confirmed by inspecting the week-by-week bias in Figure 4. What is even more striking in Figure 3 are the similarities in RMSE over time for the two affine models on the one hand and the two non-affine models on the other hand. Whether the model is affine or not is clearly much more important for its performance than whether it is formulated in continuous time or not. This confirms the message from the overall RMSEs in Table 5, but the message is much more striking when delivered visually on a week-by-week basis as in Figure 3. It is also interesting to visually inspect the relationship between the level of the volatility in the bottom panel of Figures 3 and 4 and the RMSE and bias for the different models. It is clear that in periods of high volatility, the RMSE for all models increases. Tables 6 and 7 present an analysis of the in- and out-of-sample RMSE by moneyness and maturity. The most important conclusion from Tables 6 and 7 is that the two non-affine models outperform the corresponding affine models for almost every cell in the moneyness-maturity matrix, in-sample as well as out-of-sample. Consider in particular the All strike prices rows and the All maturities columns in each panel of Tables 6 and 7. It is striking that the NA-SV model performs the best in every single case and the NA-GARCH model performs second best also in every single case. These tables also allow for some other important conclusions. For example, consider the difference between the AF-SV and AF-GARCH models. While the overall RMSE difference between the two models in Table 5 is approximately 8% in- and out-of-sample, there are important variations in the relative performance of the models across maturity. The overall RMSE of the AF-SV model is larger than that of the AF-GARCH model, but for short-maturity options the AF-SV model performs significantly better. This finding is perhaps somewhat surprising. While we believe that the limit arguments have some empirical value, our prior was that continuous-time models would prove to be somewhat restrictive because of the assumption of a continuous sample path. This restriction is well recognized in the continuous-time option valuation literature, and stochastic volatility models are augmented with jump models to improve their performance. However, jump models are believed to help the performance of stochastic volatility models mainly for short-maturity options. We therefore expected that if the Heston (1993) AF-SV model would underperform the AF-GARCH model, it would be for short-maturity options. Instead, the AF-SV model seems to underperform the AF-GARCH model mainly for longer maturities. 19

20 3.4 Conditional Moment Dynamics Ultimately, the option prices for the different models are determined by the model-implied conditional density dynamics. Therefore, we now discuss model differences by focusing on various aspects of the conditional density. In order to asses the different models ability to generate time-variation in the asymmetry of the return distribution, Figure 5 plots the conditional covariance between returns and variances for each model. Obviously, for the two stochastic volatility models, we use the discretized version of the models. We refer to this conditional covariance as the conditional leverage path, which for the four models is given by AF-GARCH : cov t (R t+1,h t+2 )= 2ach t+1 AF-SV : cov t (R t+1,v t+1 )=σρv t NA-GARCH : cov t (R t+1,h t+2 )= 2ach 3/2 t+1 NA-SV : cov t (R t+1,v t+1 )=σρv 3/2 t (3) Notice that critical differences between the affine and non-affine models show up very prominently in these conditional moments. The 3/2 term on the volatility of the non-affine models suggests that these models may be able to exhibit more variation in the conditional leverage paths. Figure 5 confirms this intuition. For each model we plot the daily conditional leverage path during , annualized by multiplying by 252. The left column uses the 199 estimates from Table 3 and the right column uses the 1995 estimates. The four rows of panels correspond to the AF- GARCH, AF-SV, NA-GARCH and NA-SV models respectively. Notice that the scaling is different between the two columns, because the 1995 estimates imply a much larger level of (and variation in) the leverage effect. The main conclusion is that for both sets of estimates the non-affine models imply more substantial leverage, as well as more substantial variation over time in the leverage effect. Given the importance of the leverage effect for option valuation, this may be a very important factor in explaining the differences in the fit betweenaffine and non-affine models documented in Table 5. Option prices are a function of the conditional variance, and therefore the variation in option prices over time is related to the conditional variance of variance. The conditional variance of variance of returns for the four models are given by AF-GARCH : Var t (h t+2 )=2a 2 +4a 2 c 2 h t+1 AF-SV : Var t (V t+1 )=σ 2 V t NA-GARCH : Var t (h t+2 )=a 2 (2 + 4c 2 )h 2 t+1 NA-SV : Var t (V t+1 )=σ 2 V 2 t (31) 2